The macroeconomics of fiscal policy and public capital

(1)

Tilburg University

The macroeconomics of fiscal policy and public capital

Duarte Bom, P.R.

Publication date:

2011

Document Version

Publisher's PDF, also known as Version of record

Link to publication in Tilburg University Research Portal

Citation for published version (APA):

Duarte Bom, P. R. (2011). The macroeconomics of fiscal policy and public capital. CentER, Center for Economic Research.

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal

Take down policy

If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

(2)

The Macroeconomics of Fiscal

Policy and Public Capital

Proefschrift

ter verkrijging van de graad van doctor aan Tilburg University op gezag van de rector magnificus, prof. dr. Ph. Eijlander, in het openbaar te verdedigen ten overstaan van een door het college voor promoties aangewezen commissie in de aula van de Universiteit op woensdag 23 november 2011 om 16.15 uur door

Pedro Ricardo Duarte Bom

(3)

(4)

(5)

(6)

Acknowledgements

The first words of gratitude must be addressed to my Ph.D. advisor, Jenny Ligthart. Jenny has supervised my research since my very first year in Tilburg, when I was still a Research Master student. She was always available to dedicate her valuable time and expertise to the success of this dissertation. Without Jenny’s support and encouragement, this dissertation would have never been possible. I am extremely indebted to her for everything I have learned and achieved in Tilburg.

I am also very thankful to Ben Heijdra, who coauthored one of the papers of this dissertation. Together with Jenny, Ben taught me almost everything I know about dynamic macroeconomics. I thank Ben and also Lex Meijdam, Monika Merz, Jan-Egbert Sturm, and Jan Jacobs for agreeing to review this dissertation. I feel deeply honored to have each of them in my Ph.D. committee.

I want to thank all the people that made my stay in Tilburg enjoyable. First, the Portuguese community. I was very fortunate to join CentER at the same time as Carlos and Geraldo. Even more fortunate did I feel when Miguel and Raposo

joined the crew one year later. Together with Tˆania, they made me feel at home

in Tilburg. Second, I’m grateful to Andrea, Bea, Chris, Gerard, Kenan, Marta, Martin, Nathanael, Patrick, Peter van der Windt, Peter van Oudheusden, Salima, Sotiris, Tunga, Vasilios, and many others, for making Tilburg feel like home. Also, I thank all my friends from Leiria and Lisbon, whose warm welcomes made home really feel like home. In particular, I gratefully acknowledge all the support I received from Catarina in all these years.

I am especially thankful to my parents and my two brothers, Jo˜ao and Tiago.

(7)

(8)

Chapter 1 Introduction and Summary

The recent economic and financial crisis brought the government to the center of the policy debate. In fact, many industrialized countries raised government expen-ditures in an attempt to stimulate economic activity. In most cases, governments prioritized investment in public infrastructure. Implicit in this choice is the belief that public investment not only stimulates aggregate demand in the short run but also expands the public capital stock, which generates positive supply-side spillovers to the private sector in the medium to long run. The dynamic effects of public investment on private sector output (and other macroeconomic variables) depend on how it affects the productivity of private capital relative to the produc-tivity of labor. On the other hand, an increase in public expenditures requires a shift of resources from the private sector to the public sector, which affects both households’ allocation of time between labor and leisure and households’ alloca-tion of disposable income between consumpalloca-tion and saving. Ultimately, changes in labor supply and private saving affect the real wage and the real interest rate, which in turn influence the levels of employment and private investment.

(13)

the long-term real interest rate effects of fiscal policy. Therefore, this chapter abandons the assumption of a small open economy facing an exogenously given rate of interest and considers instead a large economy (i.e., the United States), in which the interest rate is domestically determined. Also, compared to Chapters 3 and 4, Chapter 5 shifts the focus to the short-run effects of fiscal policy by developing a model that relates short-run interest rate changes to unexpected fiscal policy shocks. This distinction between expected and unexpected fiscal policy changes is empirically relevant in view of the quick reaction of interest rates to fiscal policy announcements.

The direct effect of public capital on output of the private sector is governed by the output elasticity of public capital, which plays a crucial role in macroeconomic models of public investment. For purposes of numerical analysis and policy evalu-ation of the macroeconomic effects of public investment, a precise measure of this parameter is required. Building on Aschauer (1989a), a fairly large body of empir-ical literature has focused on estimating the output elasticity of public capital by means of the so-called production function approach. Very little consensus on its size (and even sign) has emerged, however. Indeed, available estimates range from highly negative to strongly positive, with many statistically insignificant measure-ments falling in between. Chapter 2 of this dissertation quantitatively summarizes this literature using a sample of 578 estimates collected from 68 studies. The ob-jective of the analysis is to find the average output elasticity of public capital and uncover the determinants of excess variation around this value. We find that the ‘true’ elasticity is not homogeneous. Larger output elasticities are found in the long run, for core (rather than total) public capital and for public capital provided at a regional (rather than national) level of government. We also show that some study design characteristics—such as correction for endogeneity bias and control for spillover effects—explain part of the variation across estimates.

(14)

3 using plausible parameters taken from the literature on open economy models and from data for small open industrialized countries. We show that a permanent pub-lic investment impulse financed by lump-sum taxes crowds out private investment in both the short and the long run for the empirically plausible case of Solow-neutrality and low substitutability between private production factors. Because of the resulting lower steady-state stock of private capital, the long-run output multiplier is substantially smaller than in the standard case of a Hicks-neutral Cobb-Douglas technology. In addition, an increase in public investment improves both short-run and long-run welfare for a plausibly-sized public capital externality, despite the short-run output contraction.

Chapter 4 also studies the dynamic macroeconomic effects of public

invest-ment in a small open economy. However, the focus of this chapter is shifted

away from the production technology to the macroeconomic implications of (dis-tortionary) labor tax financing. In practice, governments do not have access to (non-distortionary) lump-sum taxes. Moreover, the recent sovereign debt crisis has severely limited the room for public debt financing in many OECD (especially European) economies. In August 2011, the political leaders of France and Ger-many called for constitutional amendments requiring eurozone nations to maintain a balanced budget. In this context, an increase in government spending must be financed by higher distortionary taxes, of which labor income taxes are an impor-tant component in industrialized countries. Therefore, a public investment impulse may generate not only a positive externality on private sector production, but its funding also distorts the labor market. We show that, for plausible parameter values, the interaction of finite life spans, the wealth effect on labor supply and the balanced budget fiscal rule gives rise to cyclical dynamic responses of output and other variables to the public investment impulse. Public investment expands output in the long run, although at the cost of an even larger short-run contrac-tion. Our numerical results suggest that, for an output elasticity of public capital of 0.08, the second-best optimal public investment ratio should be about 6 percent of GDP, which is far above the ratios currently observed for OECD countries.

(15)

has provided extremely mixed results. This chapter argues that the relationship between fiscal policy and interest rates is nonlinear. Using Markov-switching mod-els on quarterly U.S. data (1960:1-2008:4), we show that, while the effects of fiscal policy shocks are in accordance with the conventional view at times of low/declin-ing level of public debt, these effects can be reversed at times of high/rislow/declin-ing public debt. The reason is that a high stock of public debt makes households expect a high level of future taxes. If taxes are distortionary, a fiscal expansion lowers households’ lifetime income. Because households respond by strongly cutting on private consumption, the increase in private saving more than compensates the decrease in public saving, causing a decrease in the real interest rate. We show that failing to isolate these periods of ‘unconventional’ interest rate effects leads to the misleading conclusion that the Ricardian equivalence holds. Also, we show that public investment and government consumption often exert opposite effects on interest rates.

(16)

5 The chapters of this dissertation are based on the following research papers: • Chapter 2: Bom, P. R. D., and J. E. Ligthart (2008): “How Productive

is Public Capital? A Meta-Analysis,” CESifo Working Paper No. 2206, CESifo, Munich.

• Chapter 3: Bom, P. R. D., B. J. Heijdra, and J. E. Ligthart (2010): “Output Dynamics, Technology, and Public Investment,” CentER Discussion Paper No. 10-58, Tilburg University, Tilburg.

• Chapter 4: Bom, P. R. D., and J. E. Ligthart (2011): “Public Infrastructure Investment, Output Dynamics, and Balanced Budget Fiscal Rules,” CentER Discussion Paper No. 11-92, Tilburg University, Tilburg.

(17)

(18)

Chapter 2 The Output Elasticity of Public

Capital: A Meta-Analysis

(19)

2.1 Introduction

What is the quantitative effect of public capital1 _{on private output? Providing}

a solid answer to this question is of vital importance to policymakers as well as macroeconomic researchers. In the policy arena, the debate on the productivity of public capital has flared up globally following the recent world economic and financial crisis. Most industrialized countries have adopted fiscal stimulus mea-sures to address the economic crisis. Specifically, three-quarters of G-20 countries have increased public spending on infrastructure, predominantly on transportation

networks (IMF, 2009). On the research side, many endogenous growth models2

and structural macroeconomic and regional general equilibrium models feature the output elasticity of public capital as a fundamental parameter; knowledge of its exact magnitude is thus required to numerically analyze the effects of fiscal policy shocks.

The literature has devoted a great deal of effort to measuring the output elas-ticity of public capital by estimating a production function that includes public

capital as an input.3 _{Aschauer (1989a,b, 1990) uses this approach in an attempt}

to explain the productivity growth slowdown in the United States in the 1970s.4

Indeed, in the United States and various OECD countries, investments in the public capital stock fell and aggregate labor productivity growth declined slightly later. Aschauer (1989a) found that a 1 percent increase in the public capital stock increased private output by 0.39 percent, suggesting that public capital is an im-portant determinant of output. Since then, many studies have been undertaken for the United States and various other OECD countries. More recently, attention has also been focused on the productivity effects of public capital in developing countries (e.g., Ram, 1996).

Despite all these measuring efforts, remarkably little consensus has emerged in the literature. Indeed, the output elasticity of public capital differs substantially 1_{Public capital is generally defined as the tangible capital stock owned by the public sector}

excluding military structures and equipment. See Section 2.2.2.1 for a further discussion.

2_{This strand of the literature builds on the theoretical contributions of Barro (1990) and}

Glomm and Ravikumar (1994, 1997).

3_{See the literature reviews by Munnell (1991, 1992), Gramlich (1994), Pfahler, Hofmann, and}

Bonte (1996), Button (1998), Sturm, Kuper, and De Haan (1998), Button and Rietveld (2000), Mikelbank and Jackson (2000), IMF (2004), and Romp and De Haan (2007).

4_{Mera (1973) was the first to estimate a production function including some form of public}

(20)

Section 2.1 | Introduction 9

across studies, ranging from 1.7 for New Zealand to 2.04 for Australia (see

Fig-ure 2.1 below). In between these extremes, a nonnegligible share of the reported estimates are statistically not different from zero. Although the majority of es-timates are positive and cluster within a smaller range of values, it is virtually impossible to get an idea of the size of the output elasticity of public capital by glancing through the literature. The first objective of this chapter is therefore to quantify the contribution of public capital to private sector production. Our analysis shows, however, that there is no unique ‘true’ output elasticity of public capital. Estimates of the output elasticity of public capital differ across studies, not only because studies employ different data and econometric methods but also because the population effect is itself heterogeneous. Hence, the second objec-tive of the chapter is to identify the sources of heterogeneity and study design characteristics underlying the large variation of reported estimates. To this end, we apply meta-analytical techniques to a sample of primary studies. Drawing on Stanley and Jarrell (1989) and Stanley (2001), meta-analysis—broadly defined to include meta-regression analysis—can be defined as a body of statistical methods to summarize, evaluate, and analyze empirical results across studies; it presents a systematic and objective way to explain and control for the study-to-study

varia-tion.5 Our study is the first to conduct a systematic meta-regression analysis to

quantify the output elasticity of public capital.6

We focus on studies using the so-called production function approach, which includes the stock of public (infrastructure) capital as an additional input in the production function. Although other approaches exist, we focus on the production function approach because it yields a comparable measure of the effect size of

interest.7 _{The meta-sample consists of 578 estimates of the output elasticity of}

public capital collected from 68 primary studies, spanning the 1983–2008 period. We develop a meta-regression model to estimate the average output elasticity of public capital and identify the sources of variation across estimates. Because 5_{Meta-analysis has not been applied much in the public finance field. Notable exceptions are,}

amongst others, De Mooij and Ederveen (2003) and Nijkamp and Poot (2004).

6_{The studies by Button (1998) and Ligthart and Su´}_{arez (2011) also try to quantitatively}

summarize the literature on the output elasticity of public capital. However, they do not make use of meta-regression analysis (see below for a further discussion), but instead employ standard regression analysis. In addition, their analyses are rather incomplete in the coverage of the empirical literature (e.g., Button’s (1998) study consists of 26 data points) and modeling of observed heterogeneity.

7_{See Sturm et al.} _{(1998) for an overview of other approaches to estimating the output}

(21)

primary studies use samples of different sizes, the resulting estimates differ in terms of precision. To maximize the estimation efficiency of the meta-regression model, larger weights should be assigned to more precise estimates. Therefore, we estimate the meta-regression model using weighted least squares (WLS). We

consider both fixed and random effects models.8 _{However, because unobserved}

heterogeneity across observations in the meta-sample is statistically nonnegligible, the latter is preferred.

When estimating the average output elasticity of public capital, it is important to bear in mind that researchers/editors are more likely to report/publish statis-tically significant results than insignificant ones. As a result, the meta-sample may not be representative of the true population of estimates. This so-called pub-lication bias is widely recognized to inflate the size of the estimated coefficients available to the meta-analyst. Indeed, estimates obtained from small samples tend to have large standard errors. Because, in this case, only large (in absolute value) estimates are statistically significant, the meta-sample is biased toward large val-ues. Fortunately, meta-analytical techniques enable us to filter out publication bias by eliminating the correlation between estimates and standard errors (see Stanley, 2005). The meta-analysis literature typically assumes publication bias to be unidirectional. In our case, however, measurements take on both positive and negative values, suggesting that publication bias can go in both directions. For this reason, we extend the meta-regression model to allow bidirectional publication bias.

After correcting for publication bias, we find an unconditional (average) output elasticity of public capital of 0.082. The true output elasticity is rather heteroge-neous, however. In the short run, the output elasticity of public capital installed at the central level of government is only 0.051. This value is larger if primary studies consider: (i) public capital installed by regional/local governments; (ii) core public capital (i.e., roads, railways, airports, and utilities); and (iii) a long-run horizon. Conditional on these three dimensions, the output elasticity of public capital rises to 0.173. This estimate is only half the size of those found by Aschauer (1989a) and other early time-series studies, however. Because these studies make use of nonstationary data without properly testing for cointegration, our results suggest that their estimates are likely to be spurious. Moreover, we find that other study design characteristics—such as model specification, definition of variables, 8_{The meaning of the adjectives fixed and random in the meta-analysis literature differs from}

(22)

Section 2.2 | The Production Function Approach 11

and correction for endogeneity and spillover effects—explain part of the variation across estimates.

The remainder of the chapter is structured as follows. Section 2.2 presents the production function approach and discusses the key methodological issues that are raised in this literature. Section 2.3 describes the meta-sample and analyzes publi-cation bias informally. Section 2.4 sets out the meta-regression model. Section 2.5 discusses the meta-regression results. Finally, Section 2.6 concludes the chapter.

2.2 The Production Function Approach

Following the work of Aschauer (1989a), the production function approach is the most widely used in measuring the output elasticity of public capital. This section describes the empirical methodology underlying the production function approach. In addition, it discusses the main methodological issues raised in the empirical literature. The latter serves as input into defining the explanatory variables used in the meta-regression analysis of Sections 2.4 and 2.5.

2.2.1 The Empirical Model

The corner stone of the production function approach is a technological relation-ship that incorporates the stock of public capital of region/country i at time t

(denoted by Git) as an additional input:

Yit AitF pKit, Lit, Gitq , (2.1)

where Yit is aggregate private sector output of region/country i at time t, Ait is

an index of (Hicks-neutral) factor productivity,9 Kit denotes the stock of

(non-residential) private fixed capital, and Lit denotes employment (typically measured

by total hours worked). In this setup, an increase in public capital may affect

output directly (i.e., BYit{BGit ¡ 0), but also indirectly through its effect on the

marginal productivity of private factors of production (i.e., B2Yit

BKitBGit and

B2_Y it

BLitBGit).

An example of an indirect effect is the ‘crowding in’ or ‘crowding out’ of private

investment by public investment.10 _{The general idea of the production function}

9_{Note that the technology index may potentially depend on G}

it(cf. Duggal, Saltzman, and

Klein, 1999). In the context of a Cobb-Douglas production function (which is most commonly used, see below), it does not make a difference whether public capital is treated as a third input or as a factor affecting the technology index.

10_{The sign of the relationship between public capital and private capital productivity is not a}

(23)

approach is that the services of public capital—which are hypothesized to boost private output—are proportional to the stock of public capital, which is usually assumed to be a pure public good. Note that the production function approach is partial in nature; for example, the financing method of public investment is not taken into account.

For empirical purposes, a specific functional form of (2.1) is chosen. The Cobb-Douglas production function is most commonly used:

Yit AitKitαL β itG

θ

it, 0 α, β 1. (2.2)

The parameter of interest is the partial output elasticity of public capital, θ

B ln Yit

B ln Git

BYit

BGit

Git

Yit, which is typically hypothesized to be positive.

11 _{The definition}

of θ assumes that all other inputs and technology are held constant; any indirect effects of public capital on output therefore cannot be measured. Note that the Cobb-Douglas functional form imposes a unitary elasticity of substitution between factors of production, which is relaxed by more flexible functional forms like the

translog.12 _{In addition, technological progress in (2.2) is always of the}

Hicks-neutral type, that is, technology contributes equally to the productivity of all factors.

To arrive at an equation that can be estimated by linear methods, we take

natural logarithms on both sides of (2.2). Subsequently, we assume that ln Ait

a0 χt νi εit, where a0 denotes a constant, χtis a time-specific effect

(represent-ing shocks common to all units, for example, technological progress), νi denotes

an unobserved unit-specific fixed effect (e.g., the effect of climate or geographical

location on productivity), and εit is a stochastic technology shock. In its most

basic form, the regression equation is

ln Yit a0 χt νi α ln Kit β ln Lit θ ln Git εit. (2.3)

The basic model has been extended in several directions. Some studies (e.g.,

Boarnet, 1998) account for interregional spillover effects of public capital by incor-porating the capital stock of neighboring jurisdictions. Besides the basic factors of production, some studies add further controls (e.g., the business cycle, energy 11_{Traditional growth theory typically assumes diminishing returns with respect to reproducible}

factor inputs by imposing the restriction α θ 1.

12_{The transcendental logarithmic (translog) production function includes quadratic and}

(24)

prices, or education). Taking these extensions into consideration—which will be further discussed in Section 2.2.2—yields (2.3) in comprehensive form:

ln Yit a0 χt νi α ln Kit β ln Lit θ ln Git ϑ ln ˜Git γ1Zit it, (2.4)

where ˜Gitrepresents (a weighted average of) public capital of neighboring regions,

ϑ is a parameter measuring the interregional spillover effects of public capital,

Zit is vector of additional control variables, γ is the vector of coefficients of the

additional control variables, and it is an error term. Note that some studies

split total public capital into several components (see below), in which case Git

and θ—written here as scalars, for simplicity—represent vectors of public capital components and their output elasticities, respectively. As will be demonstrated below, most studies estimate a special case of (2.4).

In estimating θ, authors have employed time-series, cross-section, and panel data models. The time-series approach fixes i for one jurisdiction (typically a

country) and exploits the time variation; χtis either replaced by a linear time trend

(i.e., χt φt) or is simply not included. The seminal work of Aschauer (1989a)

analyzes time-series data and sets χt φt and ϑ γ 0. Cross-section studies

keep t fixed and exploit the variation across jurisdictions (typically states/regions;

e.g., Da Costa et al., 1987). In this context, νi cannot be estimated. Finally, panel

data models (featuring both i and t) are employed either at the regional level for a single country or at the country level for country groupings. In the panel

context, some studies treat the unobserved unit-specific effect (νi) as fixed effects

(e.g., Evans and Karras, 1994a) or as random effects (e.g., Andrews and Swanson, 1995). Alternatively, various authors use pooled ordinary least squares (OLS) so

that νi is simply ignored. A few studies (e.g., Boarnet, 1998) exclude νi by long

differencing the data.

2.2.2 Key Issues in the Literature

(25)

2.2.2.1 Defining Public Capital and Output

The Introduction provided a very general definition of public capital, which

em-phasized the government’s ownership role and the stock nature of the capital.13

More specifically, public capital consists of core infrastructure, hospitals,

educa-tional buildings, and other public buildings.14 _{Core infrastructure is generally}

perceived to be more productive than other types of public capital, such as ed-ucational and office buildings and hospitals. Indeed, studies employing a broad definition of public capital (which necessarily includes less productive components) typically find a lower output elasticity of public capital than studies focusing on core infrastructure only. Mas et al. (1994), for example, disaggregate public capi-tal in core and non-core components. Sturm and De Haan (1995) include various subcomponents of public capital into the equation all at once [see (2.4)] as well as one at a time. Some studies focus on transportation infrastructure only (e.g.,

Garcia-Mil`a and McGuire, 1992), which is a subcomponent of core infrastructure.

In countries with a fiscally decentralized government structure (e.g., the United States), different layers of government may be involved in the provision of pub-lic capital. Consequently, authors studying federal countries have consolidated their public capital stock data to differing degrees. The majority of studies em-ploy public capital stocks defined at the national level including public capital provision at all levels of government (e.g., Aschauer, 1989a), whereas others deal with capital stocks estimated for regions based on consolidated regional data (e.g.,

Garcia-Mil`a and McGuire, 1992). Some studies only consider capital that is owned

by local/regional governments (e.g., Evans and Karras, 1994a), and thus do not take into account regionally installed capital owned by the federal/central govern-ment. Because lower levels of government can better target spending on public investment, it is plausible that output elasticities are larger for this type of public capital. We test this hypothesis in Section 2.5.

The output measure used as dependent variable varies across studies. Most studies use real gross output of the private sector (e.g., Ratner, 1983) or real Gross 13_{Statistics on the public capital stock are not readily available. To arrive at an estimate}

of the public capital stock, researchers determine an initial value of the capital stock to which they add gross investment flows and subtract technical depreciation of the existing capital stock (based on the expected life spans of its components). See Sturm and De Haan (1995) for further details on this so-called perpetual inventory method.

14_{Some authors use a very broad definition of public capital by also including health and}

(26)

Domestic Product (GDP) exclusive of public sector output (e.g., Finn, 1993),

where the former does not include taxes and subsidies on products.15 Private

output may also be defined according to the economic sector where it is generated. Da Costa et al. (1987), for instance, report estimates for the manufacturing and non-agricultural sectors together with an estimate for all sectors. When the data are at the state level for the United States, real Gross State Product (GSP) is employed. Although the literature primarily deals with measuring the contribution of public capital to private output, some studies nevertheless employ a measure of total output (including public sector production). The latter is typically the case of studies using data for emerging markets or developing countries, where the only

available measure of output is total GDP (e.g., Ram, 1996).16

2.2.2.2 Returns to Scale Restrictions

Incorporating public capital into the production function raises the issue of returns to scale in production. A large number of studies impose some type of restriction on the coefficients of the production function. In a few cases, constant returns

to scale in private capital and labor is assumed (i.e., α β 1; see, e.g., Otto

and Voss, 1994), giving rise to increasing returns to scale across all inputs. If one assumes, for simplicity, that the production function is given by (2.3) (with

χt νi 0), then imposing constant returns to scale across private inputs boils

down to estimating

lnpYit{Kitq a0 β lnpLit{Kitq θ ln Git ε˜it, (2.5)

where ˜εit εit pα β 1q ln Kit. Hence, θ is inconsistently estimated if constant

returns to scale in private inputs is incorrectly assumed. If α β 1, then θ is

overestimated if private and public capital are negatively correlated. Conversely, θ is underestimated if private and public capital are positively correlated.

Aschauer (1989a) argues that congestion effects may be severe enough to render the assumption of increasing returns to scale across all inputs inappropriate. For 15_{Some studies employ (real) gross value added of the private sector (which equals net output}

of the private sector because intermediate inputs have been subtracted).

16_{Because government output is typically not exchanged on markets, it is hard to measure.}

(27)

this reason, most studies assume constant returns to scale restriction across all

inputs (i.e., α β θ 1q. To impose this restriction, (2.5) is written as

lnpYit{Kitq a0 β lnpLit{Kitq θ lnpGit{Kitq ˆεit, (2.6)

where ˆεit εit pα β θ 1q ln Kit. Therefore, θ is underestimated

(overes-timated) if constant returns to scale across all inputs is assumed while, in fact,

α β θ ¡ 1 (α β θ 1).

The question then arises as to whether output elasticities from studies (incor-rectly) imposing some type of constant returns to scale are larger or smaller than

otherwise. If the true production function is characterized by α β θ 1 and

private capital is negatively (positively) correlated with public capital, then

stud-ies imposing α β 1 overestimate (underestimate) θ. Conversely, if the true

production function features α β 1, but α β θ 1 is imposed, then θ is

underestimated. These hypotheses are empirically investigated below.

2.2.2.3 Spillover Effects of Public Capital

The first author studying the output effect of public capital in a regional context is Mera (1973), who uses very broadly defined public capital, which includes health and welfare facilities. It was not until Aschauer’s (1989a) analysis that various authors started applying his methodology to regional data using a standard

def-inition of public capital.17 They find elasticities at the regional level that tend

to be smaller than those from analyses using aggregate data for a single country. This finding can be attributed to spillover effects of public capital, that is, some

of the beneficial effects of public capital accrue to neighboring regions.18

Most studies do not include neighboring regions’ public capital stock in the home jurisdiction’s production function. The small number of studies that do (e.g., Boarnet, 1998) typically employ the weighted public capital stock of neigh-boring jurisdictions, where the weights are exogenously given. Estimating this extended equation [see (2.4)] yields an estimate of ϑ. Alternatively, some studies (e.g., Mas et al., 1994, 1996) include neighboring regions’ public capital into the

definition of Git(i.e., θ ϑ is implicitly assumed). No consensus has been reached

on the significance of spillover effects. The study by Holtz-Eakin and Schwartz 17_{Authors that take a regional approach are: Munnell (1990b), Eisner (1991), Garcia-Mil`}_{a and}

McGuire (1992), Evans and Karras (1994a), and Holtz-Eakin (1994).

18_{The theory on fiscal federalism demonstrates that in a Nash equilibrium, these spillovers}

(28)

(1995b) finds little evidence of spillover effects, whereas Boarnet (1998) obtains a significantly negative spillover coefficient. This result is not surprising because studies at this level of aggregation measure only the net effect. Backwash effects, such as congestion and resource exploitation, or displacement effects (i.e., new infrastructure shifts economic activity to other locations) may exceed any positive gross benefits of infrastructure.

2.2.2.4 Stationarity of Variables

Some of the early studies (e.g., Aschauer, 1989a,b) have been criticized for not properly accounting for stochastic trends. Generally, time series on private output, public capital stock, and other private inputs are nonstationary. If this is the case, the usual test statistics have non-standard distributions, so that standard inference procedures may give rise to misleading results (see Granger and Newbold, 1974). In particular, one may find spurious (or non-existing) relationships between the levels of private output and factor inputs.

(29)

2.2.2.5 Endogeneity Concerns

Equations (2.3) and (2.4) assume the public capital stock to be strictly exogenous, implying that the causality runs from public capital to private output. Some authors (e.g., Munnell, 1992; and Gramlich, 1994) argue that public capital is likely to be an endogenous variable. Reverse causality occurs if a higher rate of output growth boosts tax revenues, which facilitates an increase in public investment. In this case, public capital is positively correlated with the error term, causing the estimated coefficients to be upward biased. However, this positive bias may to some extent be offset if the fiscal authority follows a policy rule according to which public investment reacts anti-cyclically to changes in private output.

Some authors solve the endogeneity problem by using an Instrumental Vari-ables (IV) estimator (e.g., Holtz-Eakin, 1994; and Baltagi and Pinnoi, 1995). If the positive (negative) effect running from output to public capital dominates, then OLS estimates of θ are higher (lower) than IV estimates. Baltagi and Pinnoi (1995), for instance, use panel data for the United States to arrive at a pooled OLS estimate of θ of 0.16, whereas the IV estimate is only 0.02. Some authors (e.g., Evans and Karras, 1994a) also control for the endogeneity of other factors of pro-duction, usually labor. A few studies (e.g., Ai and Cassou, 1995) derive moment conditions from a dynamic model with optimizing firms to estimate production function parameters using the generalized method of moments (GMM) estimator. Determining the sign and size of the endogeneity bias is an empirical question, to which we turn in Section 2.5.

During the last decade, various authors have employed VAR models with a view to capturing the dynamic interactions between private output, labor, public

capital, and private capital.19 The VAR approach itself, however, does not solve

the endogeneity problem. VAR models are typically reduced-form models, in which contemporaneous effects are concentrated out. If the VAR model is to be given a structural interpretation, then the contemporaneous effects need to be uncovered. The latter requires assumptions essentially equivalent to those necessary to define appropriate instruments in an IV context.

19_{The VAR approach models every endogenous variable as a function of its own lagged value}

(30)

2.2.2.6 The Business Cycle

A number of studies (e.g., Aschauer, 1989a; Hulten and Schwab, 1991; and Sturm and De Haan, 1995) control for the effect of the business cycle on factor use by including the capital utilization rate—or, alternatively, the unemployment rate—

as an additional variable in (2.4).20 _{The effect of the capacity utilization rate}

on output is hypothesized to be positive, whereas it is negative in the case of the unemployment rate. Because authors use log-linear empirical models [recall (2.4)], capacity utilization enters the equation in an additive fashion. In some cases (e.g., Ratner, 1983), authors incorporate variables into their analysis that are pre-adjusted for business cycle effects. An alternative way of dealing with the capacity utilization issue is to employ a production frontier approach (e.g., Delorme, Thompson, and Warren, 1999). In this way, the expansionary effect of public capital on the production possibilities frontier can be disentangled from the

production-inefficiency reducing effect of public capital formation.21

Unless a long-run (cointegrating) relationship is estimated, ignoring the busi-ness cycle may lead to downward biased estimates of θ. Intuitively, public capital is perceived to be less productive during economic downturns because the

econ-omy is inside the production possibilities frontier. Technically, ln Git is negatively

correlated with εit in (2.3). This correlation vanishes in (2.4) if a measure of the

business cycle is included in Zit.

2.2.2.7 Additional Control Variables

Various authors have criticized Aschauer’s model for being misspecified due to the omission of relevant explanatory variables. Omitting variables that help explain changes in output may bias the estimate of θ if those variables are also correlated with public capital. The study by Vijverberg, Vijverberg, and Gamble (1997) proposes to add imported raw materials to equation (2.4). Some studies (e.g.,

Garcia-Mil`a and McGuire, 1992) include education as an input. Tatom (1991)

makes a case for including energy prices in the production function to account for supply shocks. For example, the rising oil prices of the 1970s—representing a neg-ative supply shock—may have depressed output and capital use. Gramlich (1994) 20_{Some studies (e.g., Garcia-Mil`}_{a and McGuire, 1992) control for the business cycle by simply}

including time fixed effects. Note that time fixed effects not only capture the business cycle but also many other time-idiosyncratic shocks.

21_{Note that in an efficient steady state, the production frontier approach is equivalent to the}

(31)

criticizes Tatom’s approach for mixing production functions and cost functions. Instead of including energy prices, which typically feature as an argument of the cost function, a measure of the quantity of energy use in production should be

employed. Most studies, however, do not include any variables in Zit.

2.3 The Meta-Sample

This section describes the meta-data set—which will be used in the regression analysis of Sections 2.4 and 2.5—and analyzes informally whether publication bias is present in the sample.

2.3.1 Constructing the Meta-Sample

Table 2.A.1 in the Appendix shows the studies that are included in our meta-data set, which covers estimates of the output elasticity of public capital based on the production function approach. In total, 68 studies are coded and included in the meta-data set. Eight papers are unpublished, whereas the remaining 60 are published in academic journals, professional journals, or books. Out of 39

published journal papers seven are published in top-20 journals.22 _{The data set}

encompasses single-country studies for 22 different countries and six cross-country analyses. In total, 30 studies (44 percent) are based on data for the United States. To obtain a sample of studies as representative as possible of the true

popu-lation of studies, we use a variety of searching methods.23 We start by checking

the references in overview papers, among others, Sturm et al. (1998) and Romp and De Haan (2007), which together provide a very comprehensive coverage of

relevant papers up to 2004.24 _{From these sources, we obtain 47 usable references}

(see below). We then search for papers citing Aschauer (1989a) in Thomson’s Web of Science, which allows us to add seven papers to our meta-data base. We also use the Internet search engine Google Scholar and search for words such as ‘public capital’ and ‘public infrastructure,’ each in combination with ‘output’ or ‘productivity,’ which yields another 14 papers (of which six are working papers).

22_{Based on the journal quality ranking of Kodrzycki and Yu (2006). Note that many studies}

in our data set do not appear on this journal list.

23_{See White (1994) for a review of the general procedures for searching and retrieving papers.} 24_{In addition, we also checked the overview papers by Pfahler et al. (1996), Button (1998),}

(32)

Section 2.3 | The Meta-Sample 21

We strive for including in the meta-sample only those studies that give rise to a comparable measure of the output elasticity of public capital. Therefore, the fol-lowing identified studies using a production function approach had to be dropped from the data base of potential studies. First, studies that use physical measures of public capital or employ public investment to proxy the public capital stock. Sec-ond, studies based on the translog production function (which do not yield a single measure of the output elasticity). Third, studies that do not distinguish between private and public infrastructure capital. Finally, studies that take a pure VAR approach, except those based on Johansen’s VAR approach to estimate parameters of cointegrated variables. Pure VAR studies are dropped from the meta-sample not only because they usually report reduced-form parameter estimates—which are different from the structural parameters of a production function—but also because the lag structure of these models gives rise to a multitude of parameter

estimates (one for each lag).25

The issue of how many estimates (also known as ‘measurements’) to include from each study (when several are reported) is still controversial in meta-analysis. Some authors (e.g., Bijmolt and Pieters, 2001) claim that all available measure-ments should be used. In contrast, Van der Sluis, Van Praag, and Vijverberg (2005) propose to select only one measurement per study and Stanley (1998, 2001) advocates to average across multiple measurements per study. We follow the first

approach and include all reported estimates in each study.26 This choice raises

two statistical problems, which will be addressed in Section 2.4. First, not all estimates in a given study are of similar statistical quality, because authors often report estimates based on ‘wrong’ models or methods to justify the choice of an

alternative (better) model or method.27 _{We solve this problem by distinguishing}

what authors explicitly consider to be non-credible (i.e., ‘wrong’) estimates from credible ones. Second, estimates both within a study and across studies that use 25_{On a more practical note, many VAR studies report impulse responses only (and thus neither}

report parameter estimates nor standard errors). The latter are used as weighting factor in the meta-analysis; see Section 2.4. Consequently, any study that does not report standard errors (e.g., Mera, 1973) had to be dropped.

26_{Bom and Ligthart (2008) follow a different approach and select only the authors’ preferred}

estimate.

27_{Take panel data studies as an example. If there are fixed effects in the data that are correlated}

(33)

Figure 2.1: Histogram of the Output Elasticity of Public Capital −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 50 100 150 200 250 � Fr equ en cy

Notes: The histogram consists of 578 estimates obtained from the 68 studies on the production function approach, which are described in Table 2.A.1 in the Appendix.

data for the same country/region are likely to be correlated. We tackle this is-sue by including a set of country dummies that control for country-specific fixed effects.

2.3.2 A Glance at the Meta-Sample

(34)

Section 2.3 | The Meta-Sample 23

Figure 2.2: Funnel Plot: All Estimates

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 100 200 300 400 500 600 � 1/se( ) �

Notes: The funnel plot considers the full sample of 578 estimates.

(35)

2.3.3 Publication Bias

Publication bias means that the sample of estimates available to the meta-analyst is not representative of the true population. This problem arises when journals are more likely to publish studies reporting statistically significant results. Papers reporting insignificant results may either not be submitted for publication (i.e., self-censoring by the author(s)) or rejected by the editors/referees (i.e., censoring by peers). As a result, available estimates are biased toward large values (in absolute value), especially in studies using small samples, for which standard errors are naturally larger.

Even though papers are not published in academic journals they may still be available as Working Papers or unpublished reports. Some authors (cf. Begg, 1994) suggest to include in the meta-sample as many unpublished studies as pos-sible with a view to minimize the perverse effects of publication bias. This strategy, however, is unable to completely eliminate publication bias. Indeed, self-censoring by authors may be quite pernicious, inducing authors from making their findings available altogether.

To informally assess whether there is publication bias in our sample, we em-ploy a second funnel plot (Figure 2.3). For ease of visualization, this funnel plot excludes the 19 most precise estimates (3 percent of the total sample) and focuses on the remaining 559 estimates. Also, we add two symmetric lines correspond-ing to negative and positive statistical significance at the 5 percent level, that is,

θ{sepθq 1.96 if θ ¡ 0, and θ{sepθq 1.96 if θ 0. In the absence of

(36)

Section 2.4 | The Meta-Regression Model 25

Figure 2.3: Funnel Plot: Most Precise Estimates Excluded

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 10 20 30 40 50 60 70 80 90 100 1/se( ) � �

Notes: The funnel plot excludes the 19 most precise estimates from the full sample of 578 estimates. Filled squares denote ‘bad’ (non-credible) estimates, while solid lines represent the relationship between θ and sepθq above which θ is statistically significant at the 5 percent level, that is, 1{sepθq 1.96{θ if θ ¡ 0 and 1{sepθq 1.96{θ if θ 0.

in both positive and negative directions.28

2.4 The Meta-Regression Model

The descriptive analysis of the meta-sample in Section 2.3 revealed not only a substantial amount of variation across estimates of the output elasticity of public capital but also some informal indications of publication bias. This section devel-ops a formal meta-regression model to estimate the average elasticity and uncover 28_{To motivate the plausibility of publication bias on the negative side, note that sampling error}

(37)

the sources of heterogeneity while correcting for publication bias. Section 2.4.1 outlines the basic model. Section 2.4.2 derives the so-called funnel asymmetry and precision effect tests. Section 2.4.3 extends the model by including observed determinants of estimate heterogeneity.

2.4.1 Heterogeneity and Publication Bias

We start by postulating a relationship between each primary estimate ˆθi and its

population parameter, which is denoted by θi:

ˆ

θi θi ςi, @ i 1, ..., N, (2.7)

where N denotes the total number of estimates in the meta-sample. In the ab-sence of publication bias, the meta-sample is representative of the population of

estimates, so that the error term ςi captures pure sampling error and is such that

Erςi|θis Epςiq 0 and V rςi|θis σςi2. Note that ςi is necessarily heteroscedastic,

because σ_ςi2 depends negatively on the sample size used to derive ˆθi.

If publication bias is present, however, the error term ςi is correlated with

any selecting factor. As discussed in Section 2.3.3, we assume that publication selection is primarily caused by statistical (in)significance. Estimates based on small samples tend to have large standard errors; because such estimates are less likely to be statistically significant, they are also less likely to be available to the

meta-analyst. As a consequence, ςi is correlated with the (reported) standard error

of ˆθi, which we denote by sepˆθiq. We assume that ςi gpsepˆθiqq µi, where µi

denotes sampling error cleansed of publication bias and satisfies Erµi|θi, sepˆθiqs

Epµiq 0 and V rµi|θi, sepˆθiqs σµi2 . Note that µi, similarly to ςi, is heteroscedastic

by definition. Taking publication bias into account, (2.7) is written as ˆ

θi θi gpsepˆθiqq µi. (2.8)

Intuitively, if the meta-sample is plagued by publication selection, then we should

observe a relationship between each estimate ˆθi and its standard error sepˆθiq. The

function gpq describes such a relationship and satisfies gp0q 0; that is, infinitely

precise estimates do not contain publication bias. Furthermore, we assume bidi-rectional publication bias, meaning that positive (negative) estimates increase

(de-crease) with the standard error, that is, g1psepˆθiqq ¡ 0 if ˆθi ¡ 0 and g1psepˆθiqq 0

if ˆθi 0. The functional form of gpq is unknown a priori. We follow Card and

(38)

assuming a linear functional form, that is, gpsepˆθiqq δsepˆθiq, where δ is a

param-eter measuring publication bias. However, because positive and negative estimates may suffer from publication bias in opposite directions—as discussed in Section 2.3.3—we modify this specification and assume the following bidirectional

publi-cation bias term: gpsepˆθiqq δpsepˆθiqPpi δnsepˆθiqPni, where Ppi(Pni) is a dummy

variable that equals one if ˆθi ¡ 0 (ˆθi 0) and zero otherwise, and δp and δn are

publication bias parameters to be estimated. Symmetric publication bias further

imposes δp δn.

We model θi as a heterogeneous parameter. In particular, we assume

θi ¯θ θnIni λi, (2.9)

where we explicitly control for non-credible estimates using the dummy variable

Ini, which equals one if estimate i is explicitly considered as senseless by the

author and zero otherwise (see Section 2.3.1). The parameter λi describes the

heterogeneity of reported estimates around ¯θ and has zero mean independently

of the realization of Ini; formally, Epλi|Iniq Epλiq 0. Hence, ¯θ denotes

the average output elasticity across credible estimates, whereas θn measures the

degree to which non-credible estimates depart from ¯θ. Substituting (2.9) and the

definition of gpq into (2.8) yields

ˆ

θi ¯θ θnIni δpsepˆθiqPpi δnsepˆθiqPni υi, (2.10)

where υi λi µi is a heteroscedastic error term consisting of a heterogeneity

component (λi) and a sampling error component (µi). In order to estimate (2.10),

we need assumptions on the heterogeneity term λi. Sections 2.4.2 and 2.4.3 discuss

a number of cases.

2.4.2 Funnel Asymmetry and Precision Effect Tests

Regarding the heterogeneity component λi, we first consider two simple models.

The simplest (but least realistic) case is the assumption of homogeneity, that is,

λi 0 for all i. Consequently, υi µi, so that Vrυi|Ini, sepˆθiqs σµi2 . This

assumption is valid if and only if all (credible) estimates are unbiased toward a

unique population effect (given by ¯θ). Following the meta-analysis literature, we

shall refer to this as a fixed effects model. The second case assumes that λi is

independent and identically distributed, with Vrλi|Ini, sepˆθiq, µis V pλiq σ2λ,

(39)

or because estimates are randomly biased toward the unique population effect ¯

θ, or a combination of both. As a result, the variance of υi contains the extra

component σ_λ2, that is, Vrυi|Ini, sepˆθiqs σµi2 σλ2. We shall refer to this as

a random effects model. In either case, (2.10) constitutes the basis of funnel asymmetry and precision effect tests, which are commonly employed in the meta-analysis literature (cf. Stanley, 2005).

Because the error term υi is heteroscedastic, we estimate (2.10) using WLS,

where the optimal weights (wi) are given by the inverse of the variance of the

error term, that is, wi 1{V rυi|Ini, sepˆθiqs. Therefore, the weights are given by

wi 1{σµi2 in the fixed effects model and wi 1{pσµi2 σλ2q in the random effects

model. To implement WLS, we need estimates of σ_µi2 and σ_λ2. A natural estimate

of σ_µi2 is sepˆθiq2, that is, the variance of ˆθi as reported in the primary studies.

For the random effects model, σ2

λ can be estimated as ˆσ2λ

NrQ{pNJq1s °N

i1wi

, where

Q, wi, and J are the residual sum of squares, estimation weights, and number of

estimated parameters, respectively, of the fixed effects model (cf. Raudenbush, 1994). To choose between fixed and random effects models, we apply the Q-test of homogeneity (cf. Shadish and Haddock, 1994), which uses the fact that the

residual sum of squares of the fixed effects model is χ2

NJ distributed under the

null hypothesis of homogeneity (cf. Raudenbush, 1994). Intuitively, this test

simply checks the statistical significance of ˆσ2_λ.

2.4.3 Observed and Unobserved Heterogeneity

This section extends the model by assuming that heterogeneity partly derives from observed study characteristics that can be captured by a set of M explanatory (so-called moderator) variables:

λi

M ¸ j1

φjDji ηi, (2.11)

where Dji is a moderator variable for study characteristic j of study i and φj

de-notes its coefficient. The error term ηi represents the remaining unobserved

hetero-geneity and is independent and identically distributed with Erηi|Dji, Ini, sepˆθiq, µis

Epηiq 0 and V rηi|Dji, Ini, sepˆθiq, µis ση2. Note also that σ2η ¤ σλ2, reflecting

that part of the heterogeneity is observed and captured by the moderator variables. Assuming (2.11), equation (2.10) generalizes to

(40)

where the new error term ξi ηi µi, similarly to υi, consists of unobserved

heterogeneity (ηi) and sampling error (µi), and satisfies Erξi|Dij, Ini, sepˆθiqs

Epξiq 0 and V rξi|Dij, Ini, sepˆθiqs ση2 σ2µi. We center all moderator variables

by removing their mean across credible estimates, so that ¯θ still represents the

average output elasticity of public capital within this group; formally, because this

procedure ensures that Epλi|Ini 0q 0, it follows from (2.9) that Epθi|Ini

0q ¯θ.

We control for observed heterogeneity by including M moderator variables in (2.12). Table 2.1 presents the complete list of variables—including their observed frequency—used in the meta-regression model. We divide the variables into several key empirical dimensions, reflecting the observable heterogeneity across measure-ments, as discussed in Section 2.2. Note that, because our list is fairly exhaustive, some potentially relevant variables (e.g., a dummy variable for panel data studies) are perfectly multicollinear with other included variables and had to be dropped. With the exception of Sample-med (i.e., the median year of the sample) and Date (i.e., the date of the study), which are continuous variables, all the remaining

variables are of the dummy type. Prior to demeaning, variables Dji of the dummy

type equal one if estimate i is described by characteristic j and zero otherwise.

As discussed in Section 2.3.1, we also include 20 country dummies29 to control for

dependency across measurements for the same country.30

We estimate (2.12) using WLS and OLS. Again, two cases regarding ηi are

considered. The first case assumes that all heterogeneity is observed, that is, ηi 0

for all i, so that σ2

η 0. This assumption gives rise to a fixed effects model, where

the optimal estimation weights are given by wi 1{σ2µi. The second case allows

for both observed and unobserved heterogeneity, that is, σ2

η ¥ 0, which gives rise to

a random effects model; the optimal weights are then given by wi 1{pσµi2 ση2q.

Parameters σ2

µi and ση2 are obtained as discussed in Section 2.4.2; the significance

of σ_η2 is tested using the Q-test outlined in the same section.

29_{Country-specific fixed effects are included for the following countries (the number of estimates}

for each country is given in parentheses): Australia (27), Belgium (3), Canada (25), Denmark (2), Finland (2), France (19), Germany (11), Greece (2), Ireland (3), Italy (34), Japan (34), the Netherlands (11), New Zealand (2), Norway (3), Portugal (15), Spain (73), Sweden (4), Switzerland (1), United Kingdom (2), and United States (278).

30_{To address within-study dependency caused by unobserved study-specific heterogeneity,}

(41)

Table 2.1: List of Moderators Used in the Meta-Regression Model Moderator (Dj) Definition N ° i1 Djia

Country fixed effect

Country k Data are for country k –

Definition of output

Private Dependent variable is gross output of the private sector 332

Manufact Dependent variable is output of the manufacturing sector 59

Agric Dependent variable is output of the agricultural sector 7

Type of public capital

Reg-gov Only public capital installed at the regional/local government

level is considered

126

Core Only core capital is used 72

Transp Only transportation capital is considered 125

Spill-agg Public capital of a given region includes public capital of

neigh-boring regions

19 Data aggregation

Reg-data Regional data are used 347

Type of data

Cross Cross-section data are used 12

Empirical model

CRTS-all Constant returns to scale on all inputs is imposed 143

CRTS-priv Constant returns to scale on private inputs is imposed 30

Pfrontier A production frontier model is employed 3

Cap-util Capacity utilization is controlled for 225

Spill-disag Public capital of neighboring areas is included as an additional

regressor

56

Energy Energy prices are controlled for 20

Education A measure of education is included as an additional input 6

Estimation method

Coint Cointegration relationship is found 44

Spurious-ts Time-series data in levels are used without cointegration test 72

Spurious-pd Panel data in levels are used without cointegration test 282

Fixed-eff Unit-specific fixed effects are employed 139

Time-eff Time-specific effects are employed 189

Trend A time trend is included in the model 165

Long-diff Equation is estimated with the variables in long differences 24

Endog Instrumental variables are employed 82

Sample

Sample-med Median year of the sample –

Publication

Published Estimate belongs to a published study 421

Date Date of the study (in years) –

(42)

Section 2.5 | Meta-Regression Results 31

2.5 Meta-Regression Results

This section presents the meta-regression results. Section 2.5.1 addresses the re-sults of the funnel asymmetry and precision effect tests, assuming both true effect homogeneity (fixed effects) and unobserved heterogeneity (random effects). Subse-quently, Section 2.5.2 controls for observed heterogeneity by extending the model to include various moderator variables.

2.5.1 Funnel Asymmetry and Precision Effect Tests

Table 2.2 reports the estimation results of equation (2.10) for both fixed effects

(FE, where σ_λ2 0) and random effects (RE, where σ_λ2 ¥ 0). For comparison

purposes, we also report estimation results using unweighted OLS. We consider three special cases of equation (2.10): (i) no correction for publication bias and bad

estimates (i.e., θn δp δn 0); (ii) bad estimates correction (i.e., δp δn 0)

only; and (iii) correction for bad estimates and publication bias (i.e., no parameter restrictions). In the latter case, we distinguish between unidirectional publication

bias (i.e., δp δn) and bidirectional publication bias (i.e., no restrictions on δp

and δn).

For the first specification, we find a simple (unweighted) average of the output elasticity of public capital of 0.188. Using WLS, however, the average elasticity drops quite substantially, while maintaining its statistical significance. This fall is particularly pronounced in the fixed effects model, in which case highly precise estimates receive a relatively larger weight. Because more precise estimates tend to be closer to zero (see Figure 2.2), the weighted average is smaller. However, the

large value of Q implies a sizable value of ˆσ2

λ, suggesting that the random effects

estimate of 0.125 is preferable.

The average effect across ‘bad’ (i.e., non-credible) estimates is significantly larger in all models. Isolating these estimates, however, causes a very small

reduc-tion in the estimate of ¯θ. In contrast, correcting for publication bias reduces the

random effects estimate of ¯θ by nearly a half, from 0.125 to 0.062. As expected, we

find strong evidence of bidirectional publication bias. However, the estimate of ¯θ

is virtually invariant to the type of publication bias. This result may simply reflect the much larger number of positive estimates than negative ones. Hence, allowing publication bias in the negative direction, although statistically significant, is not

(43)

Table 2.2: Funnel Asymmetry and Precision Effect Tests

No Correction ‘Bad’ Estimates Correction PB Correction (RE)

OLS FE RE OLS FE RE Unid Bid

¯ θ 0.188*** 0.048*** 0.125*** 0.173*** 0.045*** 0.118*** 0.062*** 0.062*** (0.013) (0.003) (0.008) (0.012) (0.003) (0.008) (0.010) (0.008) θn – – – 0.244*** 0.050 0.153*** 0.125*** 0.100*** (0.085) (0.031) (0.037) (0.035) (0.027) δ – – – – – – 1.274*** – (0.167) δp – – – – – – – 2.212*** (0.146) δn – – – – – – – -2.107*** (0.226) ¯ R2 _– _– _– _0.036 _0.036 _0.036 _0.164 _0.603 Q – – 13,927 – – 13,862 11,637 8,427 ˆ σ2 λ – – 0.006 – – 0.006 0.005 0.003

Notes: The number of observations is N 578. We report results for ordinary least squares

(OLS), fixed effects (FE), and random effects (RE) estimators. In the last two columns, ‘PB’ stands for publication bias, which can be unidirectional (‘Unid’) or bidirectional (‘Bid’). The Breusch-Godfrey test always rejects the null hypothesis of homoscedasticity at the 1 percent level (results not shown); we thus report White (heteroscedasticity-robust) standard errors in parentheses. ***, **, * denote significance at the 1, 5, and 10 percent level, respectively. Under the null hypothesis of true effect homogeneity, Q follows a χ2

N1 distribution; the p-values of

this test (not shown) are virtually zero in all cases.

2.5.2 Addressing Observed Heterogeneity

Table 2.3 reports the estimation results of the meta-regression model (2.12), which extends equation (2.10) by including the moderator variables listed in Table 2.1. We estimate the model using WLS, assuming both fixed effects and random ef-fects. For comparison purposes, we also report (unweighted) OLS estimates. The

Q-test indicates a statistically significant value of ˆσ2

η, implying that the random

effects model is more appropriate. The last column of Table 2.3 reports estimation results of a parsimonious specification that restricts the random effects model by: (i) excluding insignificant variables; (ii) aggregating variables that capture similar empirical dimensions into composite variables (see below); and (iii) imposing sym-metric publication bias. All these restrictions are easily supported by the F -test, which gives a p-value of 0.867. Therefore, we take the random effects model as our benchmark and mainly focus on its restricted specification.

The macroeconomics of fiscal policy and public capital

The Macroeconomics of Fiscal

Policy and Public Capital

Proefschrift

Pedro Ricardo Duarte Bom

Acknowledgements

Contents

Chapter 1

Introduction and Summary

Chapter 2

The Output Elasticity of Public

Capital: A Meta-Analysis

2.1

Introduction

2.2

The Production Function Approach

2.2.1

The Empirical Model

2.2.2

Key Issues in the Literature

2.3

The Meta-Sample

2.3.1

Constructing the Meta-Sample

2.3.2

A Glance at the Meta-Sample

2.3.3

Publication Bias

2.4

The Meta-Regression Model

2.4.1

Heterogeneity and Publication Bias

2.4.2

Funnel Asymmetry and Precision Effect Tests

2.4.3

Observed and Unobserved Heterogeneity

2.5

Meta-Regression Results

2.5.1

Funnel Asymmetry and Precision Effect Tests

2.5.2

Addressing Observed Heterogeneity